Rewrite as a simplified fraction. ${1.7\overline{82} = {?}}$
Solution: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1782.8282...\\ 10x &= 17.8282...\end{align*}}$ Subtracting the two equations, we have: ${990x = 1765}$ ${x = \dfrac{1765}{990}} $ ${x = \dfrac{353}{198}} \text{or} {1 \dfrac{155}{198}}$